ABSTRACT
Now we examine the Ulam-Hyers type stability for the quadratic functional equation. The following theorem is due to Skof (1983) and also independently to Cholewa (1984).
Theorem 21.1. If the function f : R→ R satisfies the inequality
exists a unique
|f(x)− q(x)| ≤ δ 2
for all x ∈ R. Proof. Let f : R→ R satisfy
|f(x+ y) + f(x− y)− 2f(x)− 2f(y)| ≤ δ (21.2) for all x, y ∈ R and some δ > 0. Letting x = 0 = y in (21.2), we see that
|f(0)| ≤ δ 2 . (21.3)
Further, letting y = x in (21.2), we see that
|f(2x) + f(0)− 4f(x)| ≤ δ, that is,
|f(2x)− 4f(x)| − |f(0)| ≤ δ which is
|f(2x)− 4f(x)| ≤ 3 2 δ (21.4)
for all x ∈ R. We replace x by 2k−1x in (21.4) to get∣∣f(2kx)− 22f(2k−1x)∣∣ ≤ 3 2 δ.
